LMFDB - Embedded newform 81.5.f.a.8.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,5,Mod(8,81)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("81.8");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	81.f (of order $18$, degree $6$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.37296700979$
Analytic rank:	$0$
Dimension:	$66$
Relative dimension:	$11$ over $\Q(\zeta_{18})$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			8.7
Character	$\chi$	$=$	81.8
Dual form			81.5.f.a.71.7

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.38785 + 0.421042i) q^{2} +(-9.51055 - 3.46156i) q^{4} +(12.2061 + 14.5467i) q^{5} +(-77.2633 + 28.1215i) q^{7} +(-54.8497 - 31.6675i) q^{8} +(23.0216 + 39.8746i) q^{10} +(-12.1711 + 14.5050i) q^{11} +(3.97248 + 22.5290i) q^{13} +(-196.333 + 34.6188i) q^{14} +(6.40988 + 5.37853i) q^{16} +(-324.171 + 187.160i) q^{17} +(-252.115 + 436.676i) q^{19} +(-65.7328 - 180.599i) q^{20} +(-35.1700 + 29.5112i) q^{22} +(244.422 - 671.544i) q^{23} +(45.9132 - 260.387i) q^{25} +55.4685i q^{26} +832.160 q^{28} +(361.455 + 63.7342i) q^{29} +(-812.225 - 295.626i) q^{31} +(664.416 + 791.820i) q^{32} +(-852.872 + 310.420i) q^{34} +(-1352.16 - 780.671i) q^{35} +(747.404 + 1294.54i) q^{37} +(-785.871 + 936.565i) q^{38} +(-208.845 - 1184.42i) q^{40} +(2056.90 - 362.687i) q^{41} +(-208.251 - 174.743i) q^{43} +(165.964 - 95.8194i) q^{44} +(866.390 - 1500.63i) q^{46} +(646.583 + 1776.47i) q^{47} +(3339.52 - 2802.19i) q^{49} +(219.267 - 602.432i) q^{50} +(40.2051 - 228.015i) q^{52} +1358.28i q^{53} -359.563 q^{55} +(5128.40 + 904.276i) q^{56} +(836.263 + 304.375i) q^{58} +(-3449.24 - 4110.64i) q^{59} +(-3419.70 + 1244.67i) q^{61} +(-1815.00 - 1047.89i) q^{62} +(1186.19 + 2054.55i) q^{64} +(-279.235 + 332.779i) q^{65} +(115.147 + 653.032i) q^{67} +(3730.90 - 657.859i) q^{68} +(-2900.06 - 2433.44i) q^{70} +(-293.218 + 169.289i) q^{71} +(-1836.71 + 3181.27i) q^{73} +(1239.63 + 3405.86i) q^{74} +(3909.33 - 3280.32i) q^{76} +(532.479 - 1462.98i) q^{77} +(-1361.74 + 7722.79i) q^{79} +158.894i q^{80} +5064.27 q^{82} +(-9503.58 - 1675.74i) q^{83} +(-6679.43 - 2431.11i) q^{85} +(-423.697 - 504.943i) q^{86} +(1126.92 - 410.165i) q^{88} +(-8338.67 - 4814.33i) q^{89} +(-940.478 - 1628.96i) q^{91} +(-4649.17 + 5540.67i) q^{92} +(795.971 + 4514.18i) q^{94} +(-9429.55 + 1662.68i) q^{95} +(-720.437 - 604.518i) q^{97} +(9154.10 - 5285.12i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} + 492 q^{11} - 6 q^{13} - 1137 q^{14} - 54 q^{16} + 9 q^{17} - 3 q^{19} + 2487 q^{20} + 1002 q^{22} + 2724 q^{23} + 435 q^{25} - 12 q^{28}+ \cdots + 259938 q^{98}+O(q^{100}) $ 66 * q + 6 * q^2 - 6 * q^4 - 3 * q^5 - 6 * q^7 + 9 * q^8 - 3 * q^10 + 492 * q^11 - 6 * q^13 - 1137 * q^14 - 54 * q^16 + 9 * q^17 - 3 * q^19 + 2487 * q^20 + 1002 * q^22 + 2724 * q^23 + 435 * q^25 - 12 * q^28 - 5016 * q^29 - 1671 * q^31 - 10224 * q^32 - 342 * q^34 + 2682 * q^35 - 3 * q^37 + 6564 * q^38 - 1113 * q^40 - 5754 * q^41 - 1266 * q^43 + 4545 * q^44 - 3 * q^46 + 9159 * q^47 + 5898 * q^49 + 34977 * q^50 + 2871 * q^52 - 12 * q^55 - 39243 * q^56 - 12291 * q^58 - 24762 * q^59 - 8358 * q^61 - 38304 * q^62 + 6141 * q^64 - 23727 * q^65 + 15996 * q^67 + 43533 * q^68 - 7251 * q^70 + 19773 * q^71 + 6108 * q^73 + 74847 * q^74 - 28614 * q^76 + 33909 * q^77 - 5658 * q^79 - 12 * q^82 - 18813 * q^83 + 24219 * q^85 - 96474 * q^86 + 36042 * q^88 - 110232 * q^89 - 6042 * q^91 - 73545 * q^92 + 2631 * q^94 + 65163 * q^95 - 3696 * q^97 + 259938 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{18}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.38785	+	0.421042i	0.596962	+	0.105260i	0.463961	−	0.885856i	$-0.346428\pi$
$2$	2.38785	+	0.421042i	0.596962	+	0.105260i	0.133000	+	0.991116i	$0.457539\pi$
$3$		0			0
$3$		0			0
$4$	−9.51055	−	3.46156i	−0.594409	−	0.216347i
$5$	12.2061	+	14.5467i	0.488245	+	0.581868i	0.952770	−	0.303692i	$-0.0982193\pi$
$5$	12.2061	+	14.5467i	0.488245	+	0.581868i	−0.464525	+	0.885560i	$0.653775\pi$
$6$		0			0
$7$	−77.2633	+	28.1215i	−1.57680	+	0.573909i	−0.974505	−	0.224364i	$-0.927970\pi$
$7$	−77.2633	+	28.1215i	−1.57680	+	0.573909i	−0.602296	+	0.798273i	$0.705747\pi$
$8$	−54.8497	−	31.6675i	−0.857026	−	0.494804i
$9$		0			0
$10$	23.0216	+	39.8746i	0.230216	+	0.398746i
$11$	−12.1711	+	14.5050i	−0.100588	+	0.119876i	−0.813990	−	0.580879i	$-0.802709\pi$
$11$	−12.1711	+	14.5050i	−0.100588	+	0.119876i	0.713402	+	0.700755i	$0.247153\pi$
$12$		0			0
$13$	3.97248	+	22.5290i	0.0235058	+	0.133308i	0.994302	−	0.106596i	$-0.0339952\pi$
$13$	3.97248	+	22.5290i	0.0235058	+	0.133308i	−0.970797	+	0.239904i	$0.922884\pi$
$14$	−196.333	+	34.6188i	−1.00170	+	0.176627i
$15$		0			0
$16$	6.40988	+	5.37853i	0.0250386	+	0.0210099i
$17$	−324.171	+	187.160i	−1.12170	+	0.647612i	−0.941834	−	0.336079i	$-0.890899\pi$
$17$	−324.171	+	187.160i	−1.12170	+	0.647612i	−0.179864	+	0.983692i	$0.557566\pi$
$18$		0			0
$19$	−252.115	+	436.676i	−0.698380	+	1.20963i	0.270648	+	0.962678i	$0.412762\pi$
$19$	−252.115	+	436.676i	−0.698380	+	1.20963i	−0.969028	+	0.246951i	$0.920571\pi$
$20$	−65.7328	−	180.599i	−0.164332	−	0.451498i
$21$		0			0
$22$	−35.1700	+	29.5112i	−0.0726653	+	0.0609735i
$23$	244.422	−	671.544i	0.462045	−	1.26946i	−0.461899	−	0.886933i	$-0.652832\pi$
$23$	244.422	−	671.544i	0.462045	−	1.26946i	0.923944	−	0.382527i	$-0.124946\pi$
$24$		0			0
$25$	45.9132	−	260.387i	0.0734611	−	0.416619i
$26$			55.4685i			0.0820540i
$27$		0			0
$28$	832.160			1.06143
$29$	361.455	+	63.7342i	0.429791	+	0.0757838i	0.384359	−	0.923184i	$-0.374422\pi$
$29$	361.455	+	63.7342i	0.429791	+	0.0757838i	0.0454321	+	0.998967i	$0.485534\pi$
$30$		0			0
$31$	−812.225	−	295.626i	−0.845187	−	0.307623i	−0.117111	−	0.993119i	$-0.537363\pi$
$31$	−812.225	−	295.626i	−0.845187	−	0.307623i	−0.728077	+	0.685496i	$0.759585\pi$
$32$	664.416	+	791.820i	0.648843	+	0.773262i
$33$		0			0
$34$	−852.872	+	310.420i	−0.737778	+	0.268529i
$35$	−1352.16	−	780.671i	−1.10381	−	0.637282i
$36$		0			0
$37$	747.404	+	1294.54i	0.545949	+	0.945612i	0.998547	+	0.0538968i	$0.0171642\pi$
$37$	747.404	+	1294.54i	0.545949	+	0.945612i	−0.452597	+	0.891715i	$0.649502\pi$
$38$	−785.871	+	936.565i	−0.544232	+	0.648591i
$39$		0			0
$40$	−208.845	−	1184.42i	−0.130528	−	0.740262i
$41$	2056.90	−	362.687i	1.22362	−	0.215757i	0.475735	−	0.879589i	$-0.342182\pi$
$41$	2056.90	−	362.687i	1.22362	−	0.215757i	0.747882	+	0.663832i	$0.231071\pi$
$42$		0			0
$43$	−208.251	−	174.743i	−0.112629	−	0.0945070i	0.584734	−	0.811225i	$-0.301199\pi$
$43$	−208.251	−	174.743i	−0.112629	−	0.0945070i	−0.697363	+	0.716718i	$0.745643\pi$
$44$	165.964	−	95.8194i	0.0857253	−	0.0494935i
$45$		0			0
$46$	866.390	−	1500.63i	0.409447	−	0.709183i
$47$	646.583	+	1776.47i	0.292704	+	0.804197i	0.995669	+	0.0929724i	$0.0296368\pi$
$47$	646.583	+	1776.47i	0.292704	+	0.804197i	−0.702965	+	0.711224i	$0.748141\pi$
$48$		0			0
$49$	3339.52	−	2802.19i	1.39089	−	1.16709i
$50$	219.267	−	602.432i	0.0877070	−	0.240973i
$51$		0			0
$52$	40.2051	−	228.015i	0.0148688	−	0.0843249i
$53$			1358.28i			0.483546i	0.970333	+	0.241773i	$0.0777289\pi$
$53$			1358.28i			0.483546i	−0.970333	+	0.241773i	$0.922271\pi$
$54$		0			0
$55$	−359.563			−0.118864
$56$	5128.40	+	904.276i	1.63533	+	0.288353i
$57$		0			0
$58$	836.263	+	304.375i	0.248592	+	0.0904800i
$59$	−3449.24	−	4110.64i	−0.990875	−	1.18088i	−0.983500	−	0.180906i	$-0.942097\pi$
$59$	−3449.24	−	4110.64i	−0.990875	−	1.18088i	−0.00737489	−	0.999973i	$-0.502348\pi$
$60$		0			0
$61$	−3419.70	+	1244.67i	−0.919026	+	0.334498i	−0.757851	−	0.652428i	$-0.773751\pi$
$61$	−3419.70	+	1244.67i	−0.919026	+	0.334498i	−0.161175	+	0.986926i	$0.551528\pi$
$62$	−1815.00	−	1047.89i	−0.472164	−	0.272604i
$63$		0			0
$64$	1186.19	+	2054.55i	0.289598	+	0.501598i
$65$	−279.235	+	332.779i	−0.0660911	+	0.0787643i
$66$		0			0
$67$	115.147	+	653.032i	0.0256510	+	0.145474i	0.994943	−	0.100437i	$-0.0320241\pi$
$67$	115.147	+	653.032i	0.0256510	+	0.145474i	−0.969292	+	0.245911i	$0.920913\pi$
$68$	3730.90	−	657.859i	0.806856	−	0.142271i
$69$		0			0
$70$	−2900.06	−	2433.44i	−0.591849	−	0.496620i
$71$	−293.218	+	169.289i	−0.0581666	+	0.0335825i	−0.528801	−	0.848746i	$-0.677358\pi$
$71$	−293.218	+	169.289i	−0.0581666	+	0.0335825i	0.470635	+	0.882328i	$0.344025\pi$
$72$		0			0
$73$	−1836.71	+	3181.27i	−0.344663	+	0.596974i	−0.985292	−	0.170876i	$-0.945340\pi$
$73$	−1836.71	+	3181.27i	−0.344663	+	0.596974i	0.640630	+	0.767850i	$0.278673\pi$
$74$	1239.63	+	3405.86i	0.226375	+	0.621961i
$75$		0			0
$76$	3909.33	−	3280.32i	0.676824	−	0.567922i
$77$	532.479	−	1462.98i	0.0898093	−	0.246749i
$78$		0			0
$79$	−1361.74	+	7722.79i	−0.218192	+	1.23743i	0.657089	+	0.753813i	$0.271788\pi$
$79$	−1361.74	+	7722.79i	−0.218192	+	1.23743i	−0.875281	+	0.483615i	$0.839323\pi$
$80$			158.894i			0.0248271i
$81$		0			0
$82$	5064.27			0.753163
$83$	−9503.58	−	1675.74i	−1.37953	−	0.243248i	−0.565825	−	0.824526i	$-0.691442\pi$
$83$	−9503.58	−	1675.74i	−1.37953	−	0.243248i	−0.813706	+	0.581277i	$0.802553\pi$
$84$		0			0
$85$	−6679.43	−	2431.11i	−0.924488	−	0.336486i
$86$	−423.697	−	504.943i	−0.0572874	−	0.0682724i
$87$		0			0
$88$	1126.92	−	410.165i	0.145522	−	0.0529655i
$89$	−8338.67	−	4814.33i	−1.05273	−	0.607793i	−0.129317	−	0.991603i	$-0.541278\pi$
$89$	−8338.67	−	4814.33i	−1.05273	−	0.607793i	−0.923412	+	0.383810i	$0.874612\pi$
$90$		0			0
$91$	−940.478	−	1628.96i	−0.113571	−	0.196710i
$92$	−4649.17	+	5540.67i	−0.549288	+	0.654616i
$93$		0			0
$94$	795.971	+	4514.18i	0.0900828	+	0.510885i
$95$	−9429.55	+	1662.68i	−1.04483	+	0.184231i
$96$		0			0
$97$	−720.437	−	604.518i	−0.0765689	−	0.0642489i	0.603700	−	0.797212i	$-0.293693\pi$
$97$	−720.437	−	604.518i	−0.0765689	−	0.0642489i	−0.680269	+	0.732963i	$0.738137\pi$
$98$	9154.10	−	5285.12i	0.953155	−	0.550304i
$99$		0			0
$100$	−1338.00	+	2317.49i	−0.133800	+	0.231749i
$101$	−203.117	−	558.059i	−0.0199115	−	0.0547063i	0.929339	−	0.369228i	$-0.120378\pi$
$101$	−203.117	−	558.059i	−0.0199115	−	0.0547063i	−0.949250	+	0.314522i	$0.898156\pi$
$102$		0			0
$103$	−5740.67	+	4816.99i	−0.541113	+	0.454048i	−0.871918	−	0.489651i	$-0.837124\pi$
$103$	−5740.67	+	4816.99i	−0.541113	+	0.454048i	0.330805	+	0.943699i	$0.392680\pi$
$104$	495.549	−	1361.51i	0.0458163	−	0.125879i
$105$		0			0
$106$	−571.893	+	3243.37i	−0.0508983	+	0.288658i
$107$			17736.2i			1.54915i	0.632483	+	0.774574i	$0.282036\pi$
$107$			17736.2i			1.54915i	−0.632483	+	0.774574i	$0.717964\pi$
$108$		0			0
$109$	−14519.8			−1.22211			−0.611053	−	0.791590i	$-0.709254\pi$
$109$	−14519.8			−1.22211			−0.611053	+	0.791590i	$0.709254\pi$
$110$	−858.580	−	151.391i	−0.0709570	−	0.0125116i
$111$		0			0
$112$	−646.501	−	235.307i	−0.0515386	−	0.0187585i
$113$	9751.79	+	11621.7i	0.763708	+	0.910152i	0.998076	−	0.0619961i	$-0.0197466\pi$
$113$	9751.79	+	11621.7i	0.763708	+	0.910152i	−0.234368	+	0.972148i	$0.575302\pi$
$114$		0			0
$115$	12752.2	−	4641.42i	0.964249	−	0.350958i
$116$	−3217.01	−	1857.34i	−0.239076	−	0.138031i
$117$		0			0
$118$	−6505.50	−	11267.9i	−0.467215	−	0.809239i
$119$	19783.3	−	23576.8i	1.39702	−	1.66491i
$120$		0			0
$121$	2480.12	+	14065.5i	0.169396	+	0.960692i
$122$	−8689.76	+	1532.24i	−0.583833	+	0.102945i
$123$		0			0
$124$	6701.38	+	5623.13i	0.435834	+	0.365708i
$125$	14626.5	−	8444.61i	0.936096	−	0.540455i
$126$		0			0
$127$	5848.51	−	10129.9i	0.362608	−	0.628056i	−0.625781	−	0.779999i	$-0.715220\pi$
$127$	5848.51	−	10129.9i	0.362608	−	0.628056i	0.988389	+	0.151943i	$0.0485530\pi$
$128$	−3689.05	−	10135.6i	−0.225162	−	0.618627i
$129$		0			0
$130$	−806.884	+	677.056i	−0.0477446	+	0.0400625i
$131$	−2357.00	+	6475.80i	−0.137346	+	0.377355i	−0.989229	−	0.146377i	$-0.953239\pi$
$131$	−2357.00	+	6475.80i	−0.137346	+	0.377355i	0.851883	+	0.523733i	$0.175461\pi$
$132$		0			0
$133$	7199.24	−	40828.9i	0.406990	−	2.30815i
$134$			1607.82i			0.0895424i
$135$		0			0
$136$	23707.5			1.28176
$137$	3706.15	+	653.494i	0.197461	+	0.0348177i	0.271504	−	0.962437i	$-0.412479\pi$
$137$	3706.15	+	653.494i	0.197461	+	0.0348177i	−0.0740428	+	0.997255i	$0.523590\pi$
$138$		0			0
$139$	6554.72	+	2385.72i	0.339254	+	0.123478i	0.506028	−	0.862517i	$-0.331113\pi$
$139$	6554.72	+	2385.72i	0.339254	+	0.123478i	−0.166775	+	0.985995i	$0.553335\pi$
$140$	10157.5	+	12105.2i	0.518238	+	0.617612i
$141$		0			0
$142$	−771.437	+	280.780i	−0.0382581	+	0.0139248i
$143$	−375.133	−	216.583i	−0.0183448	−	0.0105914i
$144$		0			0
$145$	3484.84	+	6035.92i	0.165747	+	0.287083i
$146$	−5725.23	+	6823.06i	−0.268588	+	0.320091i
$147$		0			0
$148$	−2627.09	−	14899.0i	−0.119937	−	0.680195i
$149$	13414.1	−	2365.27i	0.604213	−	0.106539i	0.136831	−	0.990594i	$-0.456308\pi$
$149$	13414.1	−	2365.27i	0.604213	−	0.106539i	0.467382	+	0.884055i	$0.345197\pi$
$150$		0			0
$151$	15756.0	+	13220.9i	0.691024	+	0.579838i	0.919204	−	0.393781i	$-0.128833\pi$
$151$	15756.0	+	13220.9i	0.691024	+	0.579838i	−0.228180	+	0.973619i	$0.573278\pi$
$152$	27656.9	−	15967.7i	1.19706	−	0.691123i
$153$		0			0
$154$	1887.45	−	3269.16i	0.0795856	−	0.137846i
$155$	−5613.75	−	15423.6i	−0.233663	−	0.641983i
$156$		0			0
$157$	−2951.50	+	2476.60i	−0.119741	+	0.100475i	−0.700692	−	0.713464i	$-0.747125\pi$
$157$	−2951.50	+	2476.60i	−0.119741	+	0.100475i	0.580951	+	0.813939i	$0.302681\pi$
$158$	−6503.24	+	17867.5i	−0.260505	+	0.715731i
$159$		0			0
$160$	−3408.42	+	19330.1i	−0.133141	+	0.755083i
$161$			58759.2i			2.26686i
$162$		0			0
$163$	−42714.3			−1.60767			−0.803837	−	0.594850i	$-0.797212\pi$
$163$	−42714.3			−1.60767			−0.803837	+	0.594850i	$0.797212\pi$
$164$	−20817.7	−	3670.72i	−0.774007	−	0.136478i
$165$		0			0
$166$	−21987.5	−	8002.81i	−0.797922	−	0.290420i
$167$	−28608.0	−	34093.7i	−1.02578	−	1.22248i	−0.974638	−	0.223788i	$-0.928158\pi$
$167$	−28608.0	−	34093.7i	−1.02578	−	1.22248i	−0.0511437	−	0.998691i	$-0.516287\pi$
$168$		0			0
$169$	26346.8	−	9589.45i	0.922474	−	0.335753i
$170$	−14925.9	−	8617.44i	−0.516465	−	0.298181i
$171$		0			0
$172$	1375.70	+	2382.78i	0.0465014	+	0.0805428i
$173$	−2355.49	+	2807.17i	−0.0787027	+	0.0937942i	−0.803959	−	0.594685i	$-0.797277\pi$
$173$	−2355.49	+	2807.17i	−0.0787027	+	0.0937942i	0.725256	+	0.688479i	$0.241721\pi$
$174$		0			0
$175$	3775.07	+	21409.5i	0.123268	+	0.699085i
$176$	−156.031	+	27.5125i	−0.00503716	+	0.000888187i
$177$		0			0
$178$	−17884.4	−	15006.8i	−0.564462	−	0.473640i
$179$	24985.1	−	14425.1i	0.779784	−	0.450209i	−0.0565697	−	0.998399i	$-0.518016\pi$
$179$	24985.1	−	14425.1i	0.779784	−	0.450209i	0.836354	+	0.548190i	$0.184683\pi$
$180$		0			0
$181$	−914.224	+	1583.48i	−0.0279059	+	0.0483344i	−0.879641	−	0.475638i	$-0.842217\pi$
$181$	−914.224	+	1583.48i	−0.0279059	+	0.0483344i	0.851735	+	0.523972i	$0.175551\pi$
$182$	−1559.86	−	4285.68i	−0.0470915	−	0.129383i
$183$		0			0
$184$	−34672.6	+	29093.7i	−1.02412	+	0.859337i
$185$	−9708.41	+	26673.6i	−0.283664	+	0.779361i
$186$		0			0
$187$	1230.77	−	6980.04i	0.0351960	−	0.199607i
$188$		−	19133.4i		−	0.541348i
$189$		0			0
$190$	−23216.4			−0.643113
$191$	31338.3	+	5525.79i	0.859032	+	0.151470i	0.585778	−	0.810472i	$-0.300789\pi$
$191$	31338.3	+	5525.79i	0.859032	+	0.151470i	0.273254	+	0.961942i	$0.411900\pi$
$192$		0			0
$193$	59142.4	+	21526.1i	1.58776	+	0.577897i	0.976873	−	0.213822i	$-0.0685911\pi$
$193$	59142.4	+	21526.1i	1.58776	+	0.577897i	0.610886	+	0.791718i	$0.290813\pi$
$194$	−1465.77	−	1746.83i	−0.0389458	−	0.0464138i
$195$		0			0
$196$	−41460.6	+	15090.4i	−1.07925	+	0.392816i
$197$	−10958.0	−	6326.62i	−0.282358	−	0.163019i	0.352133	−	0.935950i	$-0.385457\pi$
$197$	−10958.0	−	6326.62i	−0.282358	−	0.163019i	−0.634490	+	0.772931i	$0.718790\pi$
$198$		0			0
$199$	−24460.7	−	42367.2i	−0.617679	−	1.06985i	−0.989908	−	0.141710i	$-0.954740\pi$
$199$	−24460.7	−	42367.2i	−0.617679	−	1.06985i	0.372229	−	0.928141i	$-0.378593\pi$
$200$	−10764.1	+	12828.2i	−0.269103	+	0.320704i
$201$		0			0
$202$	−250.046	−	1418.08i	−0.00612797	−	0.0347535i
$203$	−29719.5	+	5240.34i	−0.721189	+	0.127165i
$204$		0			0
$205$	30382.7	+	25494.1i	0.722967	+	0.606641i
$206$	−15736.0	+	9085.18i	−0.370817	+	0.214091i
$207$		0			0
$208$	−95.7100	+	165.775i	−0.00221223	+	0.00383170i
$209$	−3265.46	−	8971.78i	−0.0747570	−	0.205393i
$210$		0			0
$211$	−761.167	+	638.695i	−0.0170968	+	0.0143459i	−0.651296	−	0.758824i	$-0.725774\pi$
$211$	−761.167	+	638.695i	−0.0170968	+	0.0143459i	0.634199	+	0.773170i	$0.281330\pi$
$212$	4701.77	−	12918.0i	0.104614	−	0.287424i
$213$		0			0
$214$	−7467.68	+	42351.3i	−0.163064	+	0.924782i
$215$		−	5162.31i		−	0.111678i
$216$		0			0
$217$	71068.6			1.50924
$218$	−34671.1	−	6113.46i	−0.729550	−	0.128639i
$219$		0			0
$220$	3419.64	+	1244.65i	0.0706537	+	0.0257158i
$221$	−5504.29	−	6559.76i	−0.112698	−	0.134309i
$222$		0			0
$223$	−35902.3	+	13067.4i	−0.721959	+	0.262772i	−0.676757	−	0.736206i	$-0.736615\pi$
$223$	−35902.3	+	13067.4i	−0.721959	+	0.262772i	−0.0452019	+	0.998978i	$0.514393\pi$
$224$	−73602.1	−	42494.2i	−1.46688	−	0.846903i
$225$		0			0
$226$	18392.5	+	31856.8i	0.360101	+	0.623714i
$227$	−36022.2	+	42929.6i	−0.699067	+	0.833115i	−0.992420	−	0.122889i	$-0.960784\pi$
$227$	−36022.2	+	42929.6i	−0.699067	+	0.833115i	0.293354	+	0.956004i	$0.405229\pi$
$228$		0			0
$229$	−12331.2	−	69933.5i	−0.235143	−	1.33356i	−0.842311	−	0.538991i	$-0.818806\pi$
$229$	−12331.2	−	69933.5i	−0.235143	−	1.33356i	0.607168	−	0.794573i	$-0.292305\pi$
$230$	32404.5	−	5713.79i	0.612562	−	0.108011i
$231$		0			0
$232$	−17807.4	−	14942.1i	−0.330844	−	0.277611i
$233$	46280.4	−	26720.0i	0.852482	−	0.492180i	−0.00900578	−	0.999959i	$-0.502867\pi$
$233$	46280.4	−	26720.0i	0.852482	−	0.492180i	0.861487	+	0.507779i	$0.169533\pi$
$234$		0			0
$235$	−17949.5	+	31089.5i	−0.325025	+	0.562960i
$236$	18574.9	+	51034.2i	0.333505	+	0.916299i
$237$		0			0
$238$	57166.2	−	47968.1i	1.00922	−	0.846835i
$239$	28076.0	−	77138.3i	0.491519	−	1.35044i	−0.407771	−	0.913084i	$-0.633694\pi$
$239$	28076.0	−	77138.3i	0.491519	−	1.35044i	0.899290	−	0.437353i	$-0.144084\pi$
$240$		0			0
$241$	−3589.80	+	20358.8i	−0.0618068	+	0.350524i	0.938184	+	0.346138i	$0.112507\pi$
$241$	−3589.80	+	20358.8i	−0.0618068	+	0.350524i	−0.999990	+	0.00438571i	$0.998604\pi$
$242$			34630.5i			0.591327i
$243$		0			0
$244$	36831.7			0.618645
$245$	81525.3	+	14375.1i	1.35819	+	0.239485i
$246$		0			0
$247$	−10839.4	−	3945.23i	−0.177669	−	0.0646663i
$248$	35188.5	+	41936.1i	0.572134	+	0.681843i
$249$		0			0
$250$	38481.4	−	14006.1i	0.615702	−	0.224097i
$251$	6936.85	+	4004.99i	0.110107	+	0.0635703i	0.554042	−	0.832489i	$-0.313085\pi$
$251$	6936.85	+	4004.99i	0.110107	+	0.0635703i	−0.443935	+	0.896059i	$0.646418\pi$
$252$		0			0
$253$	6765.85	+	11718.8i	0.105702	+	0.183080i
$254$	18230.5	−	21726.2i	0.282573	−	0.336757i
$255$		0			0
$256$	−11132.8	−	63137.0i	−0.169872	−	0.963394i
$257$	3393.10	−	598.295i	0.0513725	−	0.00905836i	−0.147903	−	0.989002i	$-0.547252\pi$
$257$	3393.10	−	598.295i	0.0513725	−	0.00905836i	0.199275	+	0.979944i	$0.436141\pi$
$258$		0			0
$259$	−94151.4	−	79002.4i	−1.40355	−	1.17772i
$260$	3807.61	−	2198.32i	0.0563256	−	0.0325196i
$261$		0			0
$262$	−8354.73	+	14470.8i	−0.121711	+	0.210810i
$263$	15704.1	+	43146.7i	0.227040	+	0.623786i	0.999942	−	0.0107384i	$-0.00341822\pi$
$263$	15704.1	+	43146.7i	0.227040	+	0.623786i	−0.772903	+	0.634524i	$0.781196\pi$
$264$		0			0
$265$	−19758.5	+	16579.4i	−0.281360	+	0.236089i
$266$	34381.3	−	94462.0i	0.485914	−	1.33504i
$267$		0			0
$268$	1165.40	−	6609.28i	0.0162257	−	0.0920206i
$269$			95674.1i			1.32218i	0.750308	+	0.661089i	$0.229905\pi$
$269$			95674.1i			1.32218i	−0.750308	+	0.661089i	$0.770095\pi$
$270$		0			0
$271$	−40434.9			−0.550577			−0.275288	−	0.961362i	$-0.588773\pi$
$271$	−40434.9			−0.550577			−0.275288	+	0.961362i	$0.588773\pi$
$272$	−3084.54	−	543.887i	−0.0416920	−	0.00735142i
$273$		0			0
$274$	8574.56	+	3120.89i	0.114212	+	0.0415697i
$275$	3218.09	+	3835.18i	0.0425533	+	0.0507131i
$276$		0			0
$277$	12974.0	−	4722.16i	0.169089	−	0.0615433i	−0.256089	−	0.966653i	$-0.582434\pi$
$277$	12974.0	−	4722.16i	0.169089	−	0.0615433i	0.425178	+	0.905110i	$0.360212\pi$
$278$	14647.2	+	8456.55i	0.189524	+	0.109422i
$279$		0			0
$280$	49443.7	+	85639.1i	0.630660	+	1.09233i
$281$	46388.8	−	55284.1i	0.587490	−	0.700144i	−0.387631	−	0.921815i	$-0.626707\pi$
$281$	46388.8	−	55284.1i	0.587490	−	0.700144i	0.975122	+	0.221671i	$0.0711510\pi$
$282$		0			0
$283$	2691.63	+	15265.0i	0.0336080	+	0.190601i	0.996990	−	0.0775329i	$-0.0247043\pi$
$283$	2691.63	+	15265.0i	0.0336080	+	0.190601i	−0.963382	+	0.268134i	$0.913593\pi$
$284$	3374.67	−	595.045i	0.0418403	−	0.00737757i
$285$		0			0
$286$	−804.570	−	675.115i	−0.00983630	−	0.00825364i
$287$	−148723.	+	85865.5i	−1.80558	+	1.04245i
$288$		0			0
$289$	28297.2	−	49012.2i	0.338803	−	0.586824i
$290$	5779.89	+	15880.1i	0.0687264	+	0.188824i
$291$		0			0
$292$	28480.3	−	23897.8i	0.334024	−	0.280280i
$293$	−19803.3	+	54409.2i	−0.230676	+	0.633778i	−0.999987	−	0.00509703i	$-0.998378\pi$
$293$	−19803.3	+	54409.2i	−0.230676	+	0.633778i	0.769311	+	0.638875i	$0.220600\pi$
$294$		0			0
$295$	17694.4	−	100350.i	0.203326	−	1.15312i
$296$		−	94673.6i		−	1.08055i
$297$		0			0
$298$	33026.8			0.371906
$299$	16100.2	+	2838.90i	0.180090	+	0.0317547i
$300$		0			0
$301$	21004.2	+	7644.91i	0.231832	+	0.0843800i
$302$	32056.5	+	38203.4i	0.351481	+	0.418879i
$303$		0			0
$304$	−3964.70	+	1443.03i	−0.0429006	+	0.0156145i
$305$	−59847.1	−	34552.7i	−0.643344	−	0.371435i
$306$		0			0
$307$	43342.8	+	75072.0i	0.459876	+	0.796528i	0.998954	−	0.0457275i	$-0.0145606\pi$
$307$	43342.8	+	75072.0i	0.459876	+	0.796528i	−0.539078	+	0.842256i	$0.681227\pi$
$308$	−10128.3	+	12070.5i	−0.106767	+	0.127240i
$309$		0			0
$310$	−6910.77	−	39192.9i	−0.0719123	−	0.407835i
$311$	75920.0	−	13386.7i	0.784938	−	0.138406i	0.233206	−	0.972427i	$-0.425078\pi$
$311$	75920.0	−	13386.7i	0.784938	−	0.138406i	0.551732	+	0.834022i	$0.313967\pi$
$312$		0			0
$313$	52397.2	+	43966.5i	0.534834	+	0.448779i	0.869767	−	0.493463i	$-0.164269\pi$
$313$	52397.2	+	43966.5i	0.534834	+	0.448779i	−0.334933	+	0.942242i	$0.608714\pi$
$314$	−8090.47	+	4671.03i	−0.0820568	+	0.0473755i
$315$		0			0
$316$	39683.7	−	68734.3i	0.397410	−	0.688334i
$317$	395.853	+	1087.60i	0.00393927	+	0.0108231i	0.941647	−	0.336603i	$-0.109278\pi$
$317$	395.853	+	1087.60i	0.00393927	+	0.0108231i	−0.937707	+	0.347426i	$0.887056\pi$
$318$		0			0
$319$	−5323.78	+	4467.18i	−0.0523165	+	0.0438987i
$320$	−15408.1	+	42333.3i	−0.150469	+	0.413411i
$321$		0			0
$322$	−24740.1	+	140308.i	−0.238610	+	1.35323i
$323$		−	188743.i		−	1.80912i
$324$		0			0
$325$	6048.65			0.0572654
$326$	−101995.	−	17984.5i	−0.959720	−	0.169224i
$327$		0			0
$328$	−124306.	−	45243.5i	−1.15543	−	0.420542i
$329$	−99914.2	−	119073.i	−0.923071	−	1.10007i
$330$		0			0
$331$	−145207.	+	52851.1i	−1.32535	+	0.482389i	−0.905170	−	0.425050i	$-0.860257\pi$
$331$	−145207.	+	52851.1i	−1.32535	+	0.482389i	−0.420184	+	0.907439i	$0.638034\pi$
$332$	84583.6	+	48834.4i	0.767379	+	0.443047i
$333$		0			0
$334$	−53956.7	−	93455.7i	−0.483674	−	0.837747i
$335$	−8093.97	+	9646.01i	−0.0721227	+	0.0859525i
$336$		0			0
$337$	20509.3	+	116314.i	0.180589	+	1.02417i	0.931493	+	0.363760i	$0.118507\pi$
$337$	20509.3	+	116314.i	0.180589	+	1.02417i	−0.750903	+	0.660412i	$0.770382\pi$
$338$	66949.6	−	11805.0i	0.586023	−	0.103332i
$339$		0			0
$340$	55109.6	+	46242.4i	0.476727	+	0.400021i
$341$	14173.8	−	8183.22i	0.121892	−	0.0703745i
$342$		0			0
$343$	−80513.1	+	139453.i	−0.684350	+	1.18533i
$344$	5888.82	+	16179.4i	0.0497636	+	0.136724i
$345$		0			0
$346$	−6806.49	+	5711.32i	−0.0568553	+	0.0477073i
$347$	−43643.0	+	119908.i	−0.362456	+	0.995839i	0.615702	+	0.787979i	$0.288872\pi$
$347$	−43643.0	+	119908.i	−0.362456	+	0.995839i	−0.978158	+	0.207861i	$0.933350\pi$
$348$		0			0
$349$	7928.22	−	44963.2i	0.0650916	−	0.369153i	−0.934810	−	0.355147i	$-0.884431\pi$
$349$	7928.22	−	44963.2i	0.0650916	−	0.369153i	0.999902	−	0.0140053i	$-0.00445818\pi$
$350$			52712.0i			0.430302i
$351$		0			0
$352$	−19572.0			−0.157961
$353$	−102037.	−	17991.9i	−0.818857	−	0.144387i	−0.251499	−	0.967858i	$-0.580923\pi$
$353$	−102037.	−	17991.9i	−0.818857	−	0.144387i	−0.567358	+	0.823471i	$0.692035\pi$
$354$		0			0
$355$	−6041.66	−	2198.98i	−0.0479402	−	0.0174488i
$356$	62640.2	+	74651.7i	0.494257	+	0.589033i
$357$		0			0
$358$	65734.1	−	23925.2i	0.512890	−	0.186677i
$359$	13349.0	+	7707.06i	0.103576	+	0.0597998i	0.550893	−	0.834576i	$-0.314287\pi$
$359$	13349.0	+	7707.06i	0.103576	+	0.0597998i	−0.447317	+	0.894375i	$0.647620\pi$
$360$		0			0
$361$	−61963.6	−	107324.i	−0.475469	−	0.823537i
$362$	−2849.74	+	3396.19i	−0.0217464	+	0.0259164i
$363$		0			0
$364$	3305.74	+	18747.8i	0.0249497	+	0.141497i
$365$	−68696.2	+	12113.0i	−0.515640	+	0.0909213i
$366$		0			0
$367$	34366.4	+	28836.8i	0.255154	+	0.214099i	0.761388	−	0.648297i	$-0.224518\pi$
$367$	34366.4	+	28836.8i	0.255154	+	0.214099i	−0.506234	+	0.862396i	$0.668963\pi$
$368$	5178.63	−	2989.88i	0.0382401	−	0.0220780i
$369$		0			0
$370$	−34412.9	+	59604.9i	−0.251373	+	0.435390i
$371$	−38196.9	−	104945.i	−0.277511	−	0.762456i
$372$		0			0
$373$	−63541.7	+	53317.8i	−0.456711	+	0.383226i	−0.841919	−	0.539604i	$-0.818574\pi$
$373$	−63541.7	+	53317.8i	−0.456711	+	0.383226i	0.385208	+	0.922830i	$0.374130\pi$
$374$	5877.78	−	16149.1i	0.0420214	−	0.115453i
$375$		0			0
$376$	20791.5	−	117914.i	0.147065	−	0.834049i
$377$			8396.41i			0.0590760i
$378$		0			0
$379$	−100679.			−0.700905			−0.350453	−	0.936581i	$-0.613972\pi$
$379$	−100679.			−0.700905			−0.350453	+	0.936581i	$0.613972\pi$
$380$	95435.7	+	16827.9i	0.660912	+	0.116537i
$381$		0			0
$382$	72504.5	+	26389.5i	0.496865	+	0.180844i
$383$	−168356.	−	200639.i	−1.14771	−	1.36779i	−0.918984	−	0.394294i	$-0.870989\pi$
$383$	−168356.	−	200639.i	−1.14771	−	1.36779i	−0.228723	−	0.973491i	$-0.573455\pi$
$384$		0			0
$385$	27781.0	−	10111.5i	0.187424	−	0.0682169i
$386$	132160.	+	76302.4i	0.887001	+	0.512110i
$387$		0			0
$388$	4759.18	+	8243.13i	0.0316132	+	0.0547556i
$389$	11245.3	−	13401.7i	0.0743145	−	0.0885646i	−0.727606	−	0.685995i	$-0.759367\pi$
$389$	11245.3	−	13401.7i	0.0743145	−	0.0885646i	0.801921	+	0.597430i	$0.203812\pi$
$390$		0			0
$391$	46451.7	+	263441.i	0.303842	+	1.72318i
$392$	−271910.	+	47945.0i	−1.76951	+	0.312012i
$393$		0			0
$394$	−23502.3	−	19720.8i	−0.151397	−	0.127037i
$395$	−128963.	+	74456.7i	−0.826552	+	0.477210i
$396$		0			0
$397$	70984.4	−	122949.i	0.450383	−	0.780086i	−0.548027	−	0.836461i	$-0.684621\pi$
$397$	70984.4	−	122949.i	0.450383	−	0.780086i	0.998410	+	0.0563749i	$0.0179542\pi$
$398$	−40570.0	−	111465.i	−0.256118	−	0.703677i
$399$		0			0
$400$	1694.80	−	1422.10i	0.0105925	−	0.00888814i
$401$	−42422.3	+	116554.i	−0.263818	+	0.724835i	0.735083	+	0.677977i	$0.237143\pi$
$401$	−42422.3	+	116554.i	−0.263818	+	0.724835i	−0.998902	+	0.0468583i	$0.985079\pi$
$402$		0			0
$403$	3433.62	−	19473.0i	0.0211418	−	0.119901i
$404$			6010.55i			0.0368257i
$405$		0			0
$406$	−73171.9			−0.443907
$407$	−27874.1	−	4914.96i	−0.168272	−	0.0296709i
$408$		0			0
$409$	−96207.7	−	35016.7i	−0.575126	−	0.209329i	0.0380487	−	0.999276i	$-0.487886\pi$
$409$	−96207.7	−	35016.7i	−0.575126	−	0.209329i	−0.613175	+	0.789947i	$0.710108\pi$
$410$	61815.1	+	73668.4i	0.367728	+	0.438241i
$411$		0			0
$412$	71271.2	−	25940.6i	0.419875	−	0.152822i
$413$	382097.	+	220604.i	2.24013	+	1.29334i
$414$		0			0
$415$	−91625.6	−	158700.i	−0.532011	−	0.921470i
$416$	−15199.6	+	18114.1i	−0.0878303	+	0.104672i
$417$		0			0
$418$	−4019.92	−	22798.1i	−0.0230073	−	0.130481i
$419$	95379.9	−	16818.0i	0.543286	−	0.0957960i	0.104732	−	0.994501i	$-0.466602\pi$
$419$	95379.9	−	16818.0i	0.543286	−	0.0957960i	0.438555	+	0.898704i	$0.355491\pi$
$420$		0			0
$421$	99854.5	+	83787.9i	0.563382	+	0.472734i	0.879442	−	0.476005i	$-0.157916\pi$
$421$	99854.5	+	83787.9i	0.563382	+	0.472734i	−0.316060	+	0.948739i	$0.602360\pi$
$422$	−2086.47	+	1204.62i	−0.0117162	+	0.00676435i
$423$		0			0
$424$	43013.3	−	74501.2i	0.239261	−	0.414412i
$425$	33850.3	+	93002.8i	0.187406	+	0.514895i
$426$		0			0
$427$	229215.	−	192334.i	1.25715	−	1.05487i
$428$	61394.8	−	168681.i	0.335154	−	0.920828i
$429$		0			0
$430$	2173.55	−	12326.8i	0.0117553	−	0.0666674i
$431$		−	304036.i		−	1.63671i	−0.574716	−	0.818353i	$-0.694887\pi$
$431$		−	304036.i		−	1.63671i	0.574716	−	0.818353i	$-0.305113\pi$
$432$		0			0
$433$	160667.			0.856942			0.428471	−	0.903555i	$-0.359052\pi$
$433$	160667.			0.856942			0.428471	+	0.903555i	$0.359052\pi$
$434$	169701.	+	29922.9i	0.900958	+	0.158863i
$435$		0			0
$436$	138092.	+	50261.2i	0.726431	+	0.264399i
$437$	231625.	+	276040.i	1.21289	+	1.44547i
$438$		0			0
$439$	180640.	−	65747.7i	0.937315	−	0.341155i	0.172210	−	0.985060i	$-0.444909\pi$
$439$	180640.	−	65747.7i	0.937315	−	0.341155i	0.765105	+	0.643905i	$0.222687\pi$
$440$	19721.9	+	11386.4i	0.101869	+	0.0588142i
$441$		0			0
$442$	−10381.5	−	17981.2i	−0.0531391	−	0.0920397i
$443$	41099.0	−	48979.9i	0.209423	−	0.249580i	−0.651100	−	0.758992i	$-0.725692\pi$
$443$	41099.0	−	48979.9i	0.209423	−	0.249580i	0.860523	+	0.509411i	$0.170137\pi$
$444$		0			0
$445$	−31750.2	−	180064.i	−0.160334	−	0.909302i
$446$	−91231.1	+	16086.5i	−0.458641	+	0.0808708i
$447$		0			0
$448$	−149426.	−	125383.i	−0.744510	−	0.624718i
$449$	287754.	−	166135.i	1.42734	−	0.824076i	0.430432	−	0.902623i	$-0.358361\pi$
$449$	287754.	−	166135.i	1.42734	−	0.824076i	0.996910	+	0.0785468i	$0.0250280\pi$
$450$		0			0
$451$	−19774.0	+	34249.6i	−0.0972170	+	0.168385i
$452$	−52515.6	−	144285.i	−0.257046	−	0.706229i
$453$		0			0
$454$	−104091.	+	87342.4i	−0.505010	+	0.423754i
$455$	12216.3	−	33564.1i	0.0590090	−	0.162126i
$456$		0			0
$457$	−50765.6	+	287906.i	−0.243073	+	1.37854i	0.581852	+	0.813295i	$0.302328\pi$
$457$	−50765.6	+	287906.i	−0.243073	+	1.37854i	−0.824925	+	0.565242i	$0.808783\pi$
$458$		−	172182.i		−	0.820838i
$459$		0			0
$460$	−137347.			−0.649088
$461$	−183042.	−	32275.2i	−0.861288	−	0.151868i	−0.274479	−	0.961593i	$-0.588505\pi$
$461$	−183042.	−	32275.2i	−0.861288	−	0.151868i	−0.586810	+	0.809725i	$0.699616\pi$
$462$		0			0
$463$	−251130.	−	91403.8i	−1.17148	−	0.426385i	−0.318297	−	0.947991i	$-0.603111\pi$
$463$	−251130.	−	91403.8i	−1.17148	−	0.426385i	−0.853187	+	0.521606i	$0.825333\pi$
$464$	1974.08	+	2352.62i	0.00916916	+	0.0109274i
$465$		0			0
$466$	121761.	−	44317.3i	0.560706	−	0.204080i
$467$	132387.	+	76433.9i	0.607034	+	0.350471i	0.771804	−	0.635861i	$-0.219355\pi$
$467$	132387.	+	76433.9i	0.607034	+	0.350471i	−0.164770	+	0.986332i	$0.552688\pi$
$468$		0			0
$469$	−27260.9	−	47217.3i	−0.123935	−	0.214662i
$470$	−55950.7	+	66679.4i	−0.253285	+	0.301853i
$471$		0			0
$472$	59015.9	+	334696.i	0.264902	+	1.50233i
$473$	5069.31	−	893.856i	0.0226583	−	0.00399526i
$474$		0			0
$475$	102129.	+	85696.7i	0.452651	+	0.379819i
$476$	−269762.	+	155747.i	−1.19060	+	0.687394i
$477$		0			0
$478$	99519.7	−	172373.i	0.435565	−	0.754421i
$479$	−37629.5	−	103386.i	−0.164005	−	0.450600i	0.830281	−	0.557344i	$-0.188180\pi$
$479$	−37629.5	−	103386.i	−0.164005	−	0.450600i	−0.994287	+	0.106744i	$0.965957\pi$
$480$		0			0
$481$	−26195.8	+	21980.9i	−0.113225	+	0.0950067i
$482$	−17143.8	+	47102.2i	−0.0737926	+	0.202743i
$483$		0			0
$484$	25101.1	−	142356.i	0.107153	−	0.607692i
$485$		−	17858.8i		−	0.0759223i
$486$		0			0
$487$	−4352.56			−0.0183521			−0.00917607	−	0.999958i	$-0.502921\pi$
$487$	−4352.56			−0.0183521			−0.00917607	+	0.999958i	$0.502921\pi$
$488$	226985.	+	40023.5i	0.953140	+	0.168064i
$489$		0			0
$490$	188617.	+	68651.1i	0.785578	+	0.285927i
$491$	97014.0	+	115617.i	0.402413	+	0.479577i	0.928754	−	0.370697i	$-0.120881\pi$
$491$	97014.0	+	115617.i	0.402413	+	0.479577i	−0.526341	+	0.850273i	$0.676437\pi$
$492$		0			0
$493$	−129101.	+	46989.1i	−0.531174	+	0.193332i
$494$	−24221.8	−	13984.4i	−0.0992549	−	0.0573048i
$495$		0			0
$496$	−3616.23	−	6263.50i	−0.0146992	−	0.0254597i
$497$	17894.3	−	21325.6i	0.0724439	−	0.0863353i
$498$		0			0
$499$	−18988.5	−	107689.i	−0.0762585	−	0.432484i	−0.998903	−	0.0468315i	$-0.985088\pi$
$499$	−18988.5	−	107689.i	−0.0762585	−	0.432484i	0.922644	−	0.385652i	$-0.126024\pi$
$500$	−168337.	+	29682.4i	−0.673350	+	0.118730i
$501$		0			0
$502$	14877.9	+	12484.0i	0.0590382	+	0.0495390i
$503$	−393566.	+	227226.i	−1.55554	+	0.898093i	−0.557868	+	0.829929i	$0.688381\pi$
$503$	−393566.	+	227226.i	−1.55554	+	0.898093i	−0.997674	+	0.0681634i	$0.978286\pi$
$504$		0			0
$505$	5638.65	−	9766.43i	0.0221102	−	0.0382960i
$506$	11221.7	+	30831.4i	0.0438286	+	0.120418i
$507$		0			0
$508$	−90687.8	+	76096.1i	−0.351416	+	0.294873i
$509$	−58931.4	+	161913.i	−0.227463	+	0.624950i	−0.999949	−	0.0100821i	$-0.996791\pi$
$509$	−58931.4	+	161913.i	−0.227463	+	0.624950i	0.772486	+	0.635032i	$0.219013\pi$
$510$		0			0
$511$	52447.9	−	297447.i	0.200857	−	1.13911i
$512$			17128.3i			0.0653394i
$513$		0			0
$514$	8354.11			0.0316209
$515$	−140143.	−	24710.9i	−0.528392	−	0.0931697i
$516$		0			0
$517$	−33637.4	−	12243.0i	−0.125846	−	0.0458043i
$518$	−191556.	−	228287.i	−0.713898	−	0.850790i
$519$		0			0
$520$	25854.2	−	9410.16i	0.0956146	−	0.0348009i
$521$	−151269.	−	87335.5i	−0.557283	−	0.321748i	0.194771	−	0.980849i	$-0.437604\pi$
$521$	−151269.	−	87335.5i	−0.557283	−	0.321748i	−0.752054	+	0.659101i	$0.770937\pi$
$522$		0			0
$523$	251302.	+	435267.i	0.918739	+	1.59130i	0.801333	+	0.598218i	$0.204124\pi$
$523$	251302.	+	435267.i	0.918739	+	1.59130i	0.117405	+	0.993084i	$0.462542\pi$
$524$	44832.7	−	53429.5i	0.163280	−	0.194589i
$525$		0			0
$526$	19332.4	+	109640.i	0.0698739	+	0.396275i
$527$	318629.	−	56182.8i	1.14726	−	0.202294i
$528$		0			0
$529$	−176859.	−	148402.i	−0.631996	−	0.530308i
$530$	−54160.9	+	31269.8i	−0.192812	+	0.111320i
$531$		0			0
$532$	−209800.	+	363385.i	−0.741281	+	1.28394i
$533$	16342.0	+	44899.2i	0.0575241	+	0.158046i
$534$		0			0
$535$	−258003.	+	216490.i	−0.901400	+	0.756364i
$536$	14364.1	−	39465.0i	0.0499975	−	0.137367i
$537$		0			0
$538$	−40282.8	+	228455.i	−0.139173	+	0.789289i
$539$			82545.6i			0.284130i
$540$		0			0
$541$	−86003.6			−0.293847			−0.146924	−	0.989148i	$-0.546937\pi$
$541$	−86003.6			−0.293847			−0.146924	+	0.989148i	$0.546937\pi$
$542$	−96552.4	−	17024.8i	−0.328673	−	0.0579540i
$543$		0			0
$544$	−363581.	−	132333.i	−1.22858	−	0.447166i
$545$	−177231.	−	211216.i	−0.596687	−	0.711104i
$546$		0			0
$547$	−62458.5	+	22733.0i	−0.208745	+	0.0759771i	−0.444277	−	0.895890i	$-0.646539\pi$
$547$	−62458.5	+	22733.0i	−0.208745	+	0.0759771i	0.235531	+	0.971867i	$0.424317\pi$
$548$	−32985.4	−	19044.1i	−0.109840	−	0.0634161i
$549$		0			0
$550$	6069.55	+	10512.8i	0.0200646	+	0.0347529i
$551$	−118959.	+	141770.i	−0.391828	+	0.466962i
$552$		0			0
$553$	−111965.	−	634982.i	−0.366126	−	2.07640i
$554$	32968.2	−	5813.18i	0.107418	−	0.0189406i
$555$		0			0
$556$	−54080.6	−	45379.1i	−0.174941	−	0.146793i
$557$	−243428.	+	140543.i	−0.784620	+	0.453001i	−0.838065	−	0.545570i	$-0.816313\pi$
$557$	−243428.	+	140543.i	−0.784620	+	0.453001i	0.0534450	+	0.998571i	$0.482980\pi$
$558$		0			0
$559$	3109.53	−	5385.86i	0.00995110	−	0.0172358i
$560$	−4468.33	−	12276.6i	−0.0142485	−	0.0391475i
$561$		0			0
$562$	134046.	−	112478.i	0.424407	−	0.356119i
$563$	41094.6	−	112907.i	0.129649	−	0.356207i	−0.857836	−	0.513924i	$-0.828191\pi$
$563$	41094.6	−	112907.i	0.129649	−	0.356207i	0.987484	+	0.157717i	$0.0504135\pi$
$564$		0			0
$565$	−50026.2	+	283713.i	−0.156711	+	0.888755i
$566$			37583.8i			0.117319i
$567$		0			0
$568$	21443.9			0.0664671
$569$	−71198.7	−	12554.2i	−0.219911	−	0.0387763i	0.0626069	−	0.998038i	$-0.480059\pi$
$569$	−71198.7	−	12554.2i	−0.219911	−	0.0387763i	−0.282518	+	0.959262i	$0.591170\pi$
$570$		0			0
$571$	430443.	+	156669.i	1.32021	+	0.480518i	0.903526	−	0.428532i	$-0.140969\pi$
$571$	430443.	+	156669.i	1.32021	+	0.480518i	0.416686	+	0.909050i	$0.363191\pi$
$572$	2818.01	+	3358.37i	0.00861292	+	0.0102645i
$573$		0			0
$574$	−391282.	+	142415.i	−1.18759	+	0.432247i
$575$	−163639.	−	94477.0i	−0.494938	−	0.285753i
$576$		0			0
$577$	212881.	+	368721.i	0.639419	+	1.10751i	0.985561	+	0.169323i	$0.0541582\pi$
$577$	212881.	+	368721.i	0.639419	+	1.10751i	−0.346142	+	0.938182i	$0.612508\pi$
$578$	88205.5	−	105119.i	0.264022	−	0.314649i
$579$		0			0
$580$	−12249.1	−	69467.9i	−0.0364122	−	0.206504i
$581$	781402.	−	137782.i	2.31485	−	0.408170i
$582$		0			0
$583$	−19701.9	−	16531.8i	−0.0579656	−	0.0486389i
$584$	201486.	−	116328.i	0.590770	−	0.341081i
$585$		0			0
$586$	−70195.8	+	121583.i	−0.204417	+	0.354060i
$587$	219405.	+	602810.i	0.636752	+	1.74946i	0.661695	+	0.749773i	$0.269837\pi$
$587$	219405.	+	602810.i	0.636752	+	1.74946i	−0.0249434	+	0.999689i	$0.507941\pi$
$588$		0			0
$589$	333867.	−	280148.i	0.962372	−	0.807526i
$590$	84503.1	−	232170.i	0.242755	−	0.666965i
$591$		0			0
$592$	−2171.96	+	12317.8i	−0.00619738	+	0.0351471i
$593$		−	446419.i		−	1.26950i	−0.772717	−	0.634751i	$-0.781103\pi$
$593$		−	446419.i		−	1.26950i	0.772717	−	0.634751i	$-0.218897\pi$
$594$		0			0
$595$	584441.			1.65085
$596$	−135763.	−	23938.7i	−0.382199	−	0.0673920i
$597$		0			0
$598$	37249.5	+	13557.7i	0.104164	+	0.0379127i
$599$	−139535.	−	166291.i	−0.388893	−	0.463464i	0.535707	−	0.844404i	$-0.320045\pi$
$599$	−139535.	−	166291.i	−0.388893	−	0.463464i	−0.924600	+	0.380939i	$0.875601\pi$
$600$		0			0
$601$	−197079.	+	71730.7i	−0.545620	+	0.198590i	−0.600100	−	0.799925i	$-0.704872\pi$
$601$	−197079.	+	71730.7i	−0.545620	+	0.198590i	0.0544792	+	0.998515i	$0.482650\pi$
$602$	46936.0	+	27098.5i	0.129513	+	0.0747744i
$603$		0			0
$604$	−104084.	−	180278.i	−0.285305	−	0.494162i
$605$	−174334.	+	207763.i	−0.476289	+	0.567619i
$606$		0			0
$607$	−37744.8	−	214061.i	−0.102442	−	0.580980i	−0.992211	−	0.124568i	$-0.960246\pi$
$607$	−37744.8	−	214061.i	−0.102442	−	0.580980i	0.889769	−	0.456412i	$-0.150866\pi$
$608$	−513278.	+	90504.8i	−1.38850	+	0.244830i
$609$		0			0
$610$	−128357.	−	107705.i	−0.344954	−	0.289451i
$611$	−37453.7	+	21623.9i	−0.100326	+	0.0579230i
$612$		0			0
$613$	−11519.4	+	19952.2i	−0.0306555	+	0.0530969i	−0.880946	−	0.473217i	$-0.843093\pi$
$613$	−11519.4	+	19952.2i	−0.0306555	+	0.0530969i	0.850291	+	0.526313i	$0.176426\pi$
$614$	71887.6	+	197510.i	0.190685	+	0.523903i
$615$		0			0
$616$	−75535.0	+	63381.4i	−0.199061	+	0.167032i
$617$	−124106.	+	340979.i	−0.326004	+	0.895689i	0.663108	+	0.748524i	$0.269237\pi$
$617$	−124106.	+	340979.i	−0.326004	+	0.895689i	−0.989112	+	0.147165i	$0.952985\pi$
$618$		0			0
$619$	2171.69	−	12316.3i	0.00566783	−	0.0321439i	−0.981843	−	0.189697i	$-0.939249\pi$
$619$	2171.69	−	12316.3i	0.00566783	−	0.0321439i	0.987511	+	0.157553i	$0.0503605\pi$
$620$			166120.i			0.432153i
$621$		0			0
$622$	186922.			0.483147
$623$	779659.	+	137475.i	2.00876	+	0.354199i
$624$		0			0
$625$	146088.	+	53171.6i	0.373985	+	0.136119i
$626$	106605.	+	127047.i	0.272037	+	0.324201i
$627$		0			0
$628$	36643.2	−	13337.0i	0.0929125	−	0.0338174i
$629$	−484573.	−	279768.i	−1.22478	−	0.707127i
$630$		0			0
$631$	−189340.	−	327946.i	−0.475535	−	0.823651i	0.524072	−	0.851674i	$-0.324412\pi$
$631$	−189340.	−	327946.i	−0.475535	−	0.823651i	−0.999607	+	0.0280228i	$0.991079\pi$
$632$	319252.	−	380470.i	0.799281	−	0.952546i
$633$		0			0
$634$	487.313	+	2763.69i	0.00121235	+	0.00687560i
$635$	218745.	−	38570.6i	0.542488	−	0.0956552i
$636$		0			0
$637$	76396.9	+	64104.6i	0.188277	+	0.157983i
$638$	−14593.2	+	8425.41i	−0.0358517	+	0.0206990i
$639$		0			0
$640$	102410.	−	177380.i	0.250025	−	0.433057i
$641$	−67858.5	−	186440.i	−0.165154	−	0.453756i	0.829316	−	0.558780i	$-0.188730\pi$
$641$	−67858.5	−	186440.i	−0.165154	−	0.453756i	−0.994470	+	0.105024i	$0.966508\pi$
$642$		0			0
$643$	266563.	−	223673.i	0.644729	−	0.540992i	−0.260737	−	0.965410i	$-0.583966\pi$
$643$	266563.	−	223673.i	0.644729	−	0.540992i	0.905466	+	0.424418i	$0.139521\pi$
$644$	203398.	−	558832.i	0.490428	−	1.34744i
$645$		0			0
$646$	79468.9	−	450690.i	0.190429	−	1.07997i
$647$			574716.i			1.37292i	0.727168	+	0.686460i	$0.240836\pi$
$647$			574716.i			1.37292i	−0.727168	+	0.686460i	$0.759164\pi$
$648$		0			0
$649$	101606.			0.241229
$650$	14443.3	+	2546.74i	0.0341852	+	0.00602778i
$651$		0			0
$652$	406236.	+	147858.i	0.955616	+	0.347816i
$653$	33785.8	+	40264.4i	0.0792334	+	0.0944266i	0.804205	−	0.594352i	$-0.202592\pi$
$653$	33785.8	+	40264.4i	0.0792334	+	0.0944266i	−0.724971	+	0.688779i	$0.758147\pi$
$654$		0			0
$655$	−122971.	+	44757.9i	−0.286630	+	0.104325i
$656$	15135.2	+	8738.31i	0.0351707	+	0.0203058i
$657$		0			0
$658$	−188445.	−	326396.i	−0.435244	−	0.753865i
$659$	50719.7	−	60445.4i	0.116790	−	0.139185i	−0.704482	−	0.709722i	$-0.748820\pi$
$659$	50719.7	−	60445.4i	0.116790	−	0.139185i	0.821272	+	0.570537i	$0.193265\pi$
$660$		0			0
$661$	−19123.0	−	108452.i	−0.0437676	−	0.248219i	0.955072	−	0.296373i	$-0.0957772\pi$
$661$	−19123.0	−	108452.i	−0.0437676	−	0.248219i	−0.998840	+	0.0481544i	$0.984666\pi$
$662$	−368985.	+	65062.0i	−0.841962	+	0.148461i
$663$		0			0
$664$	468202.	+	392868.i	1.06193	+	0.891067i
$665$	681801.	−	393638.i	1.54175	−	0.890130i
$666$		0			0
$667$	131148.	−	227155.i	0.294788	−	0.510587i
$668$	154061.	+	423278.i	0.345254	+	0.948578i
$669$		0			0
$670$	−23388.5	+	19625.3i	−0.0521019	+	0.0437187i
$671$	23567.7	−	64751.7i	0.0523446	−	0.143816i
$672$		0			0
$673$	5142.62	−	29165.2i	0.0113541	−	0.0643925i	−0.978604	−	0.205752i	$-0.934036\pi$
$673$	5142.62	−	29165.2i	0.0113541	−	0.0643925i	0.989958	+	0.141359i	$0.0451472\pi$
$674$			286376.i			0.630401i
$675$		0			0
$676$	−283767.			−0.620966
$677$	−505452.	−	89124.8i	−1.10281	−	0.194456i	−0.407531	−	0.913192i	$-0.633610\pi$
$677$	−505452.	−	89124.8i	−1.10281	−	0.194456i	−0.695283	+	0.718736i	$0.744721\pi$
$678$		0			0
$679$	72663.3	+	26447.3i	0.157607	+	0.0573643i
$680$	289377.	+	344866.i	0.625816	+	0.745818i
$681$		0			0
$682$	37290.2	−	13572.5i	0.0801727	−	0.0291805i
$683$	211439.	+	122074.i	0.453256	+	0.261687i	0.709204	−	0.705003i	$-0.249054\pi$
$683$	211439.	+	122074.i	0.453256	+	0.261687i	−0.255948	+	0.966690i	$0.582388\pi$
$684$		0			0
$685$	35731.6	+	61888.9i	0.0761501	+	0.131896i
$686$	−250968.	+	299092.i	−0.533299	+	0.635561i
$687$		0			0
$688$	−395.002	−	2240.17i	−0.000834493	−	0.00473264i
$689$	−30600.8	+	5395.74i	−0.0644605	+	0.0113661i
$690$		0			0
$691$	−216628.	−	181773.i	−0.453689	−	0.380691i	0.387113	−	0.922032i	$-0.373472\pi$
$691$	−216628.	−	181773.i	−0.453689	−	0.380691i	−0.840803	+	0.541342i	$0.817917\pi$
$692$	32119.2	−	18544.0i	0.0670737	−	0.0387250i
$693$		0			0
$694$	−154699.	+	267946.i	−0.321195	+	0.556326i
$695$	45303.4	+	124470.i	0.0937910	+	0.257689i
$696$		0			0
$697$	−598906.	+	502542.i	−1.23280	+	1.03444i
$698$	37862.7	−	104027.i	0.0777143	−	0.213518i
$699$		0			0
$700$	38207.2	−	216684.i	0.0779738	−	0.442211i
$701$		−	448128.i		−	0.911941i	−0.889995	−	0.455970i	$-0.849292\pi$
$701$		−	448128.i		−	0.911941i	0.889995	−	0.455970i	$-0.150708\pi$
$702$		0			0
$703$	−753728.			−1.52512
$704$	−44238.5	−	7800.45i	−0.0892597	−	0.0157389i
$705$		0			0
$706$	−236073.	−	85923.7i	−0.473628	−	0.172387i
$707$	31387.0	+	37405.5i	0.0627929	+	0.0748336i
$708$		0			0
$709$	−457909.	+	166665.i	−0.910934	+	0.331553i	−0.754626	−	0.656156i	$-0.772181\pi$
$709$	−457909.	+	166665.i	−0.910934	+	0.331553i	−0.156308	+	0.987708i	$0.549959\pi$
$710$	−13500.7	−	7794.63i	−0.0267818	−	0.0154625i
$711$		0			0
$712$	304915.	+	528129.i	0.601477	+	1.04179i
$713$	−397051.	+	473187.i	−0.781030	+	0.930795i
$714$		0			0
$715$	−1428.35	−	8100.60i	−0.00279398	−	0.0158455i
$716$	−287555.	+	50703.7i	−0.560912	+	0.0989040i
$717$		0			0
$718$	28630.4	+	24023.8i	0.0555365	+	0.0466007i
$719$	377380.	−	217880.i	0.729997	−	0.421464i	−0.0884242	−	0.996083i	$-0.528183\pi$
$719$	377380.	−	217880.i	0.729997	−	0.421464i	0.818421	+	0.574619i	$0.194850\pi$
$720$		0			0
$721$	308082.	−	533613.i	0.592646	−	1.02649i
$722$	−102772.	−	282363.i	−0.197151	−	0.541668i
$723$		0			0
$724$	14176.1	−	11895.1i	0.0270445	−	0.0226930i
$725$	33191.1	−	91191.7i	0.0631459	−	0.173492i
$726$		0			0
$727$	−147103.	+	834264.i	−0.278326	+	1.57846i	0.449869	+	0.893095i	$0.351471\pi$
$727$	−147103.	+	834264.i	−0.278326	+	1.57846i	−0.728195	+	0.685370i	$0.759640\pi$
$728$			119130.i			0.224781i
$729$		0			0
$730$	−169136.			−0.317388
$731$	100214.	+	17670.4i	0.187540	+	0.0330683i
$732$		0			0
$733$	−695767.	−	253239.i	−1.29496	−	0.471326i	−0.399607	−	0.916686i	$-0.630853\pi$
$733$	−695767.	−	253239.i	−1.29496	−	0.471326i	−0.895352	+	0.445360i	$0.853076\pi$
$734$	69920.2	+	83327.6i	0.129781	+	0.154667i
$735$		0			0
$736$	694140.	−	252646.i	1.28142	−	0.466398i
$737$	−10873.7	−	6277.94i	−0.0200190	−	0.0115580i
$738$		0			0
$739$	222419.	+	385242.i	0.407271	+	0.705414i	0.994583	−	0.103946i	$-0.0331471\pi$
$739$	222419.	+	385242.i	0.407271	+	0.705414i	−0.587312	+	0.809361i	$0.699814\pi$
$740$	184665.	−	220075.i	0.337225	−	0.401889i
$741$		0			0
$742$	−47022.1	−	266676.i	−0.0854072	−	0.484368i
$743$	34938.0	−	6160.52i	0.0632879	−	0.0111594i	−0.141915	−	0.989879i	$-0.545326\pi$
$743$	34938.0	−	6160.52i	0.0632879	−	0.0111594i	0.205202	+	0.978720i	$0.434215\pi$
$744$		0			0
$745$	198142.	+	166261.i	0.356996	+	0.299555i
$746$	−174177.	+	100561.i	−0.312977	+	0.180698i
$747$		0			0
$748$	−35867.1	+	62123.7i	−0.0641052	+	0.111033i
$749$	−498769.	−	1.37036e6i	−0.889070	−	2.44270i
$750$		0			0
$751$	152855.	−	128260.i	0.271019	−	0.227412i	−0.497141	−	0.867670i	$-0.665617\pi$
$751$	152855.	−	128260.i	0.271019	−	0.227412i	0.768160	+	0.640258i	$0.221173\pi$
$752$	−5410.28	+	14864.6i	−0.00956718	+	0.0262856i
$753$		0			0
$754$	−3535.24	+	20049.3i	−0.00621836	+	0.0352661i
$755$			390574.i			0.685188i
$756$		0			0
$757$	548038.			0.956354			0.478177	−	0.878264i	$-0.341298\pi$
$757$	548038.			0.956354			0.478177	+	0.878264i	$0.341298\pi$
$758$	−240405.	−	42389.9i	−0.418413	−	0.0737776i
$759$		0			0
$760$	569861.	+	207412.i	0.986601	+	0.359093i
$761$	−200787.	−	239289.i	−0.346711	−	0.413194i	0.564304	−	0.825567i	$-0.309144\pi$
$761$	−200787.	−	239289.i	−0.346711	−	0.413194i	−0.911015	+	0.412373i	$0.864700\pi$
$762$		0			0
$763$	1.12185e6	−	408320.i	1.92702	−	0.701377i
$764$	−278917.	−	161033.i	−0.477846	−	0.275885i
$765$		0			0
$766$	−317531.	−	549980.i	−0.541164	−	0.937323i
$767$	78906.8	−	94037.4i	0.134129	−	0.159849i
$768$		0			0
$769$	151598.	+	859755.i	0.256354	+	1.45386i	0.792572	+	0.609778i	$0.208741\pi$
$769$	151598.	+	859755.i	0.256354	+	1.45386i	−0.536218	+	0.844080i	$0.680147\pi$
$770$	70594.1	−	12447.6i	0.119066	−	0.0209945i
$771$		0			0
$772$	−487963.	−	409450.i	−0.818752	−	0.687015i
$773$	325414.	−	187878.i	0.544599	−	0.314425i	−0.202342	−	0.979315i	$-0.564855\pi$
$773$	325414.	−	187878.i	0.544599	−	0.314425i	0.746941	+	0.664890i	$0.231522\pi$
$774$		0			0
$775$	−114269.	+	197920.i	−0.190250	+	0.329523i
$776$	20372.2	+	55972.0i	0.0338309	+	0.0929496i
$777$		0			0
$778$	32494.8	−	27266.4i	0.0536853	−	0.0450473i
$779$	−360199.	+	989638.i	−0.593564	+	1.63080i
$780$		0			0
$781$	1113.25	−	6313.57i	0.00182512	−	0.0103508i
$782$			648614.i			1.06065i
$783$		0			0
$784$	36477.6			0.0593463
$785$	−72052.7	−	12704.8i	−0.116926	−	0.0206172i
$786$		0			0
$787$	30613.3	+	11142.3i	0.0494266	+	0.0179898i	0.366615	−	0.930373i	$-0.380517\pi$
$787$	30613.3	+	11142.3i	0.0494266	+	0.0179898i	−0.317189	+	0.948362i	$0.602739\pi$
$788$	82316.9	+	98101.4i	0.132567	+	0.157988i
$789$		0			0
$790$	−339293.	+	123492.i	−0.543651	+	0.197873i
$791$	−1.08028e6	−	623698.i	−1.72656	−	0.996830i
$792$		0			0
$793$	−41625.8	−	72098.1i	−0.0661937	−	0.114651i
$794$	221266.	−	263695.i	0.350973	−	0.418274i
$795$		0			0
$796$	85978.3	+	487607.i	0.135695	+	0.769562i
$797$	−1.18058e6	+	208169.i	−1.85858	+	0.327717i	−0.986769	−	0.162133i	$-0.948163\pi$
$797$	−1.18058e6	+	208169.i	−1.85858	+	0.327717i	−0.871807	+	0.489850i	$0.837051\pi$
$798$		0			0
$799$	−542087.	−	454865.i	−0.849133	−	0.712507i
$800$	236685.	−	136650.i	0.369820	−	0.213516i
$801$		0			0
$802$	−150372.	+	260452.i	−0.233786	+	0.404929i
$803$	−23789.5	−	65361.2i	−0.0368939	−	0.101365i
$804$		0			0
$805$	−854753.	+	717223.i	−1.31901	+	1.10678i
$806$	16397.9	−	45052.9i	0.0252417	−	0.0693510i
$807$		0			0
$808$	−6531.42	+	37041.5i	−0.0100043	+	0.0567370i
$809$			1.08016e6i			1.65041i	0.564834	+	0.825205i	$0.308940\pi$
$809$			1.08016e6i			1.65041i	−0.564834	+	0.825205i	$0.691060\pi$
$810$		0			0
$811$	−874301.			−1.32929			−0.664644	−	0.747160i	$-0.731417\pi$
$811$	−874301.			−1.32929			−0.664644	+	0.747160i	$0.731417\pi$
$812$	300788.	+	53037.1i	0.456193	+	0.0804391i
$813$		0			0
$814$	−64489.7	−	23472.3i	−0.0973288	−	0.0354248i
$815$	−521376.	−	621352.i	−0.784939	−	0.935454i
$816$		0			0
$817$	128810.	−	46882.9i	0.192976	−	0.0702377i
$818$	−214986.	−	124122.i	−0.321294	−	0.185499i
$819$		0			0
$820$	−200707.	−	347634.i	−0.298493	−	0.517005i
$821$	529901.	−	631511.i	0.786155	−	0.936903i	−0.213039	−	0.977044i	$-0.568336\pi$
$821$	529901.	−	631511.i	0.786155	−	0.936903i	0.999194	+	0.0401408i	$0.0127806\pi$
$822$		0			0
$823$	174771.	+	991177.i	0.258030	+	1.46336i	0.788174	+	0.615453i	$0.211027\pi$
$823$	174771.	+	991177.i	0.258030	+	1.46336i	−0.530144	+	0.847908i	$0.677862\pi$
$824$	467416.	−	82418.0i	0.688413	−	0.121386i
$825$		0			0
$826$	819505.	+	687647.i	1.20113	+	1.00787i
$827$	707410.	−	408423.i	1.03433	−	0.597172i	0.116109	−	0.993236i	$-0.462958\pi$
$827$	707410.	−	408423.i	1.03433	−	0.597172i	0.918223	+	0.396065i	$0.129624\pi$
$828$		0			0
$829$	−385454.	+	667625.i	−0.560871	+	0.971457i	0.436550	+	0.899680i	$0.356200\pi$
$829$	−385454.	+	667625.i	−0.560871	+	0.971457i	−0.997421	+	0.0717770i	$0.977133\pi$
$830$	−151968.	−	417530.i	−0.220596	−	0.606082i
$831$		0			0
$832$	−41574.9	+	34885.5i	−0.0600598	+	0.0503962i
$833$	−558116.	+	1.53341e6i	−0.804331	+	2.20988i
$834$		0			0
$835$	146758.	−	832305.i	0.210488	−	1.19374i
$836$			96630.1i			0.138261i
$837$		0			0
$838$	234834.			0.334405
$839$	53061.4	+	9356.16i	0.0753798	+	0.0132915i	0.211211	−	0.977441i	$-0.432259\pi$
$839$	53061.4	+	9356.16i	0.0753798	+	0.0132915i	−0.135831	+	0.990732i	$0.543370\pi$
$840$		0			0
$841$	−538039.	−	195830.i	−0.760715	−	0.276878i
$842$	203159.	+	242115.i	0.286558	+	0.341506i
$843$		0			0
$844$	9450.00	−	3439.52i	0.0132662	−	0.00482851i
$845$	461087.	+	266209.i	0.645758	+	0.372828i
$846$		0			0
$847$	−587166.	−	1.01700e6i	−0.818453	−	1.41760i
$848$	−7305.55	+	8706.42i	−0.0101592	+	0.0121073i
$849$		0			0
$850$	41671.1	+	236329.i	0.0576763	+	0.327099i
$851$	1.05202e6	−	185500.i	1.45267	−	0.256145i
$852$		0			0
$853$	369011.	+	309637.i	0.507155	+	0.425554i	0.860127	−	0.510080i	$-0.170384\pi$
$853$	369011.	+	309637.i	0.507155	+	0.425554i	−0.352972	+	0.935634i	$0.614829\pi$
$854$	628311.	−	362755.i	0.861507	−	0.497391i
$855$		0			0
$856$	561660.	−	972824.i	0.766525	−	1.32766i
$857$	−125149.	−	343845.i	−0.170399	−	0.468167i	0.824871	−	0.565322i	$-0.191248\pi$
$857$	−125149.	−	343845.i	−0.170399	−	0.468167i	−0.995269	+	0.0971550i	$0.969026\pi$
$858$		0			0
$859$	660472.	−	554202.i	0.895093	−	0.751072i	−0.0741321	−	0.997248i	$-0.523619\pi$
$859$	660472.	−	554202.i	0.895093	−	0.751072i	0.969225	+	0.246176i	$0.0791742\pi$
$860$	−17869.6	+	49096.4i	−0.0241612	+	0.0663824i
$861$		0			0
$862$	128012.	−	725992.i	0.172280	−	0.977051i
$863$			98163.6i			0.131804i	0.997826	+	0.0659021i	$0.0209925\pi$
$863$			98163.6i			0.131804i	−0.997826	+	0.0659021i	$0.979008\pi$
$864$		0			0
$865$	−69586.5			−0.0930021
$866$	383649.	+	67647.6i	0.511562	+	0.0902021i
$867$		0			0
$868$	−675901.	−	246008.i	−0.897106	−	0.326520i
$869$	−95445.2	−	113747.i	−0.126391	−	0.150626i
$870$		0			0
$871$	−14254.8	+	5188.31i	−0.0187899	+	0.00683896i
$872$	796408.	+	459806.i	1.04738	+	0.604703i
$873$		0			0
$874$	436860.	+	756664.i	0.571899	+	0.990559i
$875$	−892615.	+	1.06378e6i	−1.16587	+	1.38942i
$876$		0			0
$877$	−228699.	−	1.29702e6i	−0.297348	−	1.68635i	−0.657501	−	0.753453i	$-0.728387\pi$
$877$	−228699.	−	1.29702e6i	−0.297348	−	1.68635i	0.360153	−	0.932893i	$-0.382725\pi$
$878$	459024.	−	80938.3i	0.595451	−	0.104994i
$879$		0			0
$880$	−2304.75	−	1933.92i	−0.00297618	−	0.00249731i
$881$	−449905.	+	259753.i	−0.579655	+	0.334664i	−0.760996	−	0.648756i	$-0.775289\pi$
$881$	−449905.	+	259753.i	−0.579655	+	0.334664i	0.181342	+	0.983420i	$0.441956\pi$
$882$		0			0
$883$	−169836.	+	294164.i	−0.217825	+	0.377284i	−0.954143	−	0.299352i	$-0.903230\pi$
$883$	−169836.	+	294164.i	−0.217825	+	0.377284i	0.736318	+	0.676636i	$0.236563\pi$
$884$	29641.9	+	81440.4i	0.0379316	+	0.104216i
$885$		0			0
$886$	118761.	−	99652.1i	0.151288	−	0.126946i
$887$	467802.	−	1.28528e6i	0.594586	−	1.63361i	−0.167303	−	0.985906i	$-0.553506\pi$
$887$	467802.	−	1.28528e6i	0.594586	−	1.63361i	0.761889	−	0.647707i	$-0.224272\pi$
$888$		0			0
$889$	−167006.	+	947140.i	−0.211314	+	1.19842i
$890$		−	443335.i		−	0.559695i
$891$		0			0
$892$	386684.			0.485989
$893$	−938756.	−	165528.i	−1.17720	−	0.207572i
$894$		0			0
$895$	514809.	+	187375.i	0.642688	+	0.233919i
$896$	570057.	+	679367.i	0.710072	+	0.846230i
$897$		0			0
$898$	757061.	−	275548.i	0.938811	−	0.341699i
$899$	−274741.	−	158622.i	−0.339941	−	0.196265i
$900$		0			0
$901$	−254216.	−	440315.i	−0.313150	−	0.542392i
$902$	−61637.9	+	73457.2i	−0.0757591	+	0.0902862i
$903$		0			0
$904$	−166852.	−	946262.i	−0.204171	−	1.15791i
$905$	−34193.6	+	6029.25i	−0.0417491	+	0.00736150i
$906$		0			0
$907$	−224759.	−	188595.i	−0.273213	−	0.229253i	0.495878	−	0.868392i	$-0.334846\pi$
$907$	−224759.	−	188595.i	−0.273213	−	0.229253i	−0.769091	+	0.639139i	$0.779291\pi$
$908$	491194.	−	283591.i	0.595774	−	0.343970i
$909$		0			0
$910$	43302.6	−	75002.3i	0.0522915	−	0.0905716i
$911$	402064.	+	1.10466e6i	0.484461	+	1.33104i	0.905632	+	0.424064i	$0.139397\pi$
$911$	402064.	+	1.10466e6i	0.484461	+	1.33104i	−0.421172	+	0.906981i	$0.638381\pi$
$912$		0			0
$913$	139976.	−	117454.i	0.167924	−	0.140905i
$914$	−242441.	+	666101.i	−0.290211	+	0.797347i
$915$		0			0
$916$	−124803.	+	707790.i	−0.148742	+	0.843556i
$917$		−	566624.i		−	0.673839i
$918$		0			0
$919$	945391.			1.11939			0.559694	−	0.828700i	$-0.310919\pi$
$919$	945391.			1.11939			0.559694	+	0.828700i	$0.310919\pi$
$920$	−846436.	−	149249.i	−1.00004	−	0.176334i
$921$		0			0
$922$	−423487.	−	154137.i	−0.498170	−	0.181319i
$923$	−4978.73	−	5933.42i	−0.00584407	−	0.00696469i
$924$		0			0
$925$	371397.	−	135178.i	0.434066	−	0.157987i
$926$	−561175.	−	323994.i	−0.654449	−	0.377847i
$927$		0			0
$928$	189690.	+	328553.i	0.220267	+	0.381513i
$929$	960369.	−	1.14452e6i	1.11277	−	1.32615i	0.172779	−	0.984961i	$-0.444725\pi$
$929$	960369.	−	1.14452e6i	1.11277	−	1.32615i	0.939994	−	0.341190i	$-0.110830\pi$
$930$		0			0
$931$	381706.	+	2.16476e6i	0.440382	+	2.49753i
$932$	−532644.	+	93919.6i	−0.613205	+	0.108125i
$933$		0			0
$934$	283939.	+	238253.i	0.325485	+	0.273114i
$935$	116560.	−	67295.7i	0.133329	−	0.0769776i
$936$		0			0
$937$	624225.	−	1.08119e6i	0.710988	−	1.23147i	−0.253499	−	0.967336i	$-0.581582\pi$
$937$	624225.	−	1.08119e6i	0.710988	−	1.23147i	0.964487	−	0.264131i	$-0.0850852\pi$
$938$	−45214.4	−	124226.i	−0.0513892	−	0.141191i
$939$		0			0
$940$	278328.	−	233545.i	0.314993	−	0.264311i
$941$	261870.	−	719483.i	0.295738	−	0.812534i	−0.699462	−	0.714670i	$-0.746577\pi$
$941$	261870.	−	719483.i	0.295738	−	0.812534i	0.995200	−	0.0978637i	$-0.0312009\pi$
$942$		0			0
$943$	259191.	−	1.46995e6i	0.291472	−	1.65302i
$944$		−	44900.5i		−	0.0503857i
$945$		0			0
$946$	12481.1			0.0139467
$947$	1.48819e6	+	262408.i	1.65943	+	0.292602i	0.923254	−	0.384189i	$-0.125519\pi$
$947$	1.48819e6	+	262408.i	1.65943	+	0.292602i	0.736175	+	0.676791i	$0.236630\pi$
$948$		0			0
$949$	−78967.3	−	28741.8i	−0.0876829	−	0.0319140i
$950$	207787.	+	247631.i	0.230235	+	0.274384i
$951$		0			0
$952$	−1.83172e6	+	666692.i	−2.02109	+	0.735616i
$953$	624485.	+	360547.i	0.687601	+	0.396987i	0.802713	−	0.596366i	$-0.203389\pi$
$953$	624485.	+	360547.i	0.687601	+	0.396987i	−0.115112	+	0.993353i	$0.536723\pi$
$954$		0			0
$955$	302138.	+	523318.i	0.331282	+	0.573798i
$956$	−534037.	+	636441.i	−0.584327	+	0.696373i
$957$		0			0
$958$	−46323.5	−	262714.i	−0.0504744	−	0.286254i
$959$	−304726.	+	53731.5i	−0.331339	+	0.0584240i
$960$		0			0
$961$	−135143.	−	113399.i	−0.146335	−	0.122789i
$962$	−71806.3	+	41457.4i	−0.0775912	+	0.0447973i
$963$		0			0
$964$	104614.	−	181197.i	0.112573	−	0.194983i
$965$	408767.	+	1.12308e6i	0.438956	+	1.20602i
$966$		0			0
$967$	875922.	−	734986.i	0.936726	−	0.786006i	−0.0402869	−	0.999188i	$-0.512827\pi$
$967$	875922.	−	734986.i	0.936726	−	0.786006i	0.977013	+	0.213182i	$0.0683827\pi$
$968$	309384.	−	850026.i	0.330178	−	0.907155i
$969$		0			0
$970$	7519.31	−	42644.1i	0.00799161	−	0.0453227i
$971$			1.26624e6i			1.34300i	0.741003	+	0.671502i	$0.234351\pi$
$971$			1.26624e6i			1.34300i	−0.741003	+	0.671502i	$0.765649\pi$
$972$		0			0
$973$	−573529.			−0.605801
$974$	−10393.2	−	1832.61i	−0.0109555	−	0.00193175i
$975$		0			0
$976$	−28614.3	−	10414.8i	−0.0300389	−	0.0109333i
$977$	−458620.	−	546562.i	−0.480467	−	0.572598i	0.470299	−	0.882507i	$-0.344146\pi$
$977$	−458620.	−	546562.i	−0.480467	−	0.572598i	−0.950766	+	0.309909i	$0.899701\pi$
$978$		0			0
$979$	171323.	−	62356.5i	0.178752	−	0.0650603i
$980$	−725590.	−	418919.i	−0.755508	−	0.436193i
$981$		0			0
$982$	182975.	+	316922.i	0.189744	+	0.328647i
$983$	57552.1	−	68587.9i	0.0595599	−	0.0709807i	−0.735442	−	0.677588i	$-0.763025\pi$
$983$	57552.1	−	68587.9i	0.0595599	−	0.0709807i	0.795002	+	0.606607i	$0.207470\pi$
$984$		0			0
$985$	−41723.7	−	236627.i	−0.0430041	−	0.243888i
$986$	−328059.	+	57845.6i	−0.337441	+	0.0594999i
$987$		0			0
$988$	89432.2	+	75042.5i	0.0916178	+	0.0768765i
$989$	−168249.	+	97138.7i	−0.172013	+	0.0993115i
$990$		0			0
$991$	−854306.	+	1.47970e6i	−0.869894	+	1.50670i	−0.00778888	+	0.999970i	$0.502479\pi$
$991$	−854306.	+	1.47970e6i	−0.869894	+	1.50670i	−0.862105	+	0.506730i	$0.830854\pi$
$992$	−305573.	−	839554.i	−0.310521	−	0.853150i
$993$		0			0
$994$	51707.8	−	43388.0i	0.0523339	−	0.0439134i
$995$	317732.	−	872962.i	0.320933	−	0.881758i
$996$		0			0
$997$	−71080.3	+	403116.i	−0.0715087	+	0.405546i	0.927952	+	0.372700i	$0.121568\pi$
$997$	−71080.3	+	403116.i	−0.0715087	+	0.405546i	−0.999460	+	0.0328455i	$0.989543\pi$
$998$		−	265139.i		−	0.266203i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.5.f.a.8.7		66
3.2	odd	2		27.5.f.a.2.5	✓	66
27.13	even	9		27.5.f.a.14.5	yes	66
27.14	odd	18	inner	81.5.f.a.71.7		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.2.5	✓	66	3.2	odd	2
27.5.f.a.14.5	yes	66	27.13	even	9
81.5.f.a.8.7		66	1.1	even	1	trivial
81.5.f.a.71.7		66	27.14	odd	18	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	81.f (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.38785 + 0.421042i) q^{2} +(-9.51055 - 3.46156i) q^{4} +(12.2061 + 14.5467i) q^{5} +(-77.2633 + 28.1215i) q^{7} +(-54.8497 - 31.6675i) q^{8} +(23.0216 + 39.8746i) q^{10} +(-12.1711 + 14.5050i) q^{11} +(3.97248 + 22.5290i) q^{13} +(-196.333 + 34.6188i) q^{14} +(6.40988 + 5.37853i) q^{16} +(-324.171 + 187.160i) q^{17} +(-252.115 + 436.676i) q^{19} +(-65.7328 - 180.599i) q^{20} +(-35.1700 + 29.5112i) q^{22} +(244.422 - 671.544i) q^{23} +(45.9132 - 260.387i) q^{25} +55.4685i q^{26} +832.160 q^{28} +(361.455 + 63.7342i) q^{29} +(-812.225 - 295.626i) q^{31} +(664.416 + 791.820i) q^{32} +(-852.872 + 310.420i) q^{34} +(-1352.16 - 780.671i) q^{35} +(747.404 + 1294.54i) q^{37} +(-785.871 + 936.565i) q^{38} +(-208.845 - 1184.42i) q^{40} +(2056.90 - 362.687i) q^{41} +(-208.251 - 174.743i) q^{43} +(165.964 - 95.8194i) q^{44} +(866.390 - 1500.63i) q^{46} +(646.583 + 1776.47i) q^{47} +(3339.52 - 2802.19i) q^{49} +(219.267 - 602.432i) q^{50} +(40.2051 - 228.015i) q^{52} +1358.28i q^{53} -359.563 q^{55} +(5128.40 + 904.276i) q^{56} +(836.263 + 304.375i) q^{58} +(-3449.24 - 4110.64i) q^{59} +(-3419.70 + 1244.67i) q^{61} +(-1815.00 - 1047.89i) q^{62} +(1186.19 + 2054.55i) q^{64} +(-279.235 + 332.779i) q^{65} +(115.147 + 653.032i) q^{67} +(3730.90 - 657.859i) q^{68} +(-2900.06 - 2433.44i) q^{70} +(-293.218 + 169.289i) q^{71} +(-1836.71 + 3181.27i) q^{73} +(1239.63 + 3405.86i) q^{74} +(3909.33 - 3280.32i) q^{76} +(532.479 - 1462.98i) q^{77} +(-1361.74 + 7722.79i) q^{79} +158.894i q^{80} +5064.27 q^{82} +(-9503.58 - 1675.74i) q^{83} +(-6679.43 - 2431.11i) q^{85} +(-423.697 - 504.943i) q^{86} +(1126.92 - 410.165i) q^{88} +(-8338.67 - 4814.33i) q^{89} +(-940.478 - 1628.96i) q^{91} +(-4649.17 + 5540.67i) q^{92} +(795.971 + 4514.18i) q^{94} +(-9429.55 + 1662.68i) q^{95} +(-720.437 - 604.518i) q^{97} +(9154.10 - 5285.12i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 6 q^{2} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} + 492 q^{11} - 6 q^{13} - 1137 q^{14} - 54 q^{16} + 9 q^{17} - 3 q^{19} + 2487 q^{20} + 1002 q^{22} + 2724 q^{23} + 435 q^{25} - 12 q^{28}+ \cdots + 259938 q^{98}+O(q^{100}) \) 66 * q + 6 * q^2 - 6 * q^4 - 3 * q^5 - 6 * q^7 + 9 * q^8 - 3 * q^10 + 492 * q^11 - 6 * q^13 - 1137 * q^14 - 54 * q^16 + 9 * q^17 - 3 * q^19 + 2487 * q^20 + 1002 * q^22 + 2724 * q^23 + 435 * q^25 - 12 * q^28 - 5016 * q^29 - 1671 * q^31 - 10224 * q^32 - 342 * q^34 + 2682 * q^35 - 3 * q^37 + 6564 * q^38 - 1113 * q^40 - 5754 * q^41 - 1266 * q^43 + 4545 * q^44 - 3 * q^46 + 9159 * q^47 + 5898 * q^49 + 34977 * q^50 + 2871 * q^52 - 12 * q^55 - 39243 * q^56 - 12291 * q^58 - 24762 * q^59 - 8358 * q^61 - 38304 * q^62 + 6141 * q^64 - 23727 * q^65 + 15996 * q^67 + 43533 * q^68 - 7251 * q^70 + 19773 * q^71 + 6108 * q^73 + 74847 * q^74 - 28614 * q^76 + 33909 * q^77 - 5658 * q^79 - 12 * q^82 - 18813 * q^83 + 24219 * q^85 - 96474 * q^86 + 36042 * q^88 - 110232 * q^89 - 6042 * q^91 - 73545 * q^92 + 2631 * q^94 + 65163 * q^95 - 3696 * q^97 + 259938 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more